We are only beginning to understand what the meaning of the, “phenomenological velocity,” solution truly is and how the curvatures that result from the solutions to the v-variable are effecting the perceived phenomena in our reality

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Mathematical Analysis in Ecological Optics: Mathematical, Theoretical Applications to The Work of James J. GibsonI. IntroductionEcological optics is the theory of visual perception introduced by James J. Gibson in his 1966 work, The Senses Considered as Perceptual Systems1 and developed the theory further in his 1979 work, The Ecological Approach to Visual Perception2. Gibson s perceptual theory developed over the years, 1947-1979 that he was doing academic research, but I plan to return to his earlier ideas, because I feel like there are themes within his 1950 work, The Perception of the Visual World3 that are still valid approaches to problems of perception and perceptual phenomena. Use of eidetic phenomenology (a discussion of which is present in the paper, Overview of Perceptual Theories (Emmerson, 2010)) helps understand the ideas of surface layout and gradient, because it can be used to deliver actual surfaces with layout, texture, mathematical and physical significance, and contour. Use of1 Gibson, James J.. The Senses Considered as Perceptual Systems. London: George Allen & Unwin Ltd, 1966. Print. All further references to this source will be cited parenthetically in the text.2 Gibson, James J.. The Ecological Approach To Visual Perception. 1 ed. Hillside, NJ: Tf- Lea, 1986. Print. All further references to this source will be cited parenthetically in the text.3 Gibson, James, J.. The Perception of the Visual World. Cambridge, Mass.: The Riverside Press, 1950. Print. All further references to this source will be cited parenthetically in the text. 1eidetic phenomenology helps understand the implications of formal ontology of Gestalt theory and interpretation of Pragnanz, or pithiness, which tells us that the visual system organizes information in a simple, systematic, and symmetric manner4. The parameters of the formally ontological system contain many gradients that describe a distance from the eye to a point out in the world. Gibson says, the problem of how distance can be perceived is very old. If it is taken to be the distance of an object in space, then it is a line endwise to the eye, as Bishop Berkley pointed out in 1709, and it projects only one point on the retina. Hence, distance of itself is invisible and, if so, a whole set of perplexities arise that have never been resolved (Ecological Approach, 117).The formally ontological system proposed in The Geometric Pattern of Perception Theorems (Emmerson, 2009), is proposed as a way of resolving some of these perplexities, and is a whole system. I will be using Gibson s work, The Ecological Approach to Visual Perception in order to describe the ideas of ambient optic array, surface layout, optical information, and the role of geometry in perceptual studies.This paper will use the gradient of the mathematics of difference in circumferences equaling an arc length transformed through the Pythagorean theorem to form a cone to describe surface layouts as examples of how mathematics can be a useful language for describing perceived layout; whereas, the kind of information that single shot photographs contain is not useful. This is not to say anything against photography as art, because often, the photograph makes us feel4 Koffka, K.. Principles of Gestalt Psychology. First Edition ed. London: Kegan Paul, Trench, Trubner, 1935. Print. 2present to the artist s vision, personality or message. The Geometric Pattern of Perception Theorems (Emmerson, 2009) exhibit the qualities of Pragnanz, because they contain symmetry, and are formally, ontologically based through mathematical, simple, description of the phenomenon of perceived change. The Gestalt theory will be connected to the theory of Ecological Optics, because a, univocal (it speaks with one voice through geometric difference) system, with the qualities of Pragnanz, describes the form of the ambient optic array through transcendental logic involving computation. I use the term transcendental logic to correlate to ideality as an extension of Husserl s idea, transcendence belonging to the real, as such, is a particular form of ideality or, better, of a psychic irreality: the irreality of something that itself, with all that belongs to it in its own essence, actually or possibly makes its appearance in the purely phenomenological sphere of consciousness. 5 I will draw several parallels from ideas in Gibson s work to my own work.The results of visualizing solutions to the equations can be discussed through the language of visual gradients, surface layouts, textures, and contours, which Gibson introduced in philosophically organized conversation in his 1950 work, The Perception of the Visual World. Gibson s theory evolved dramatically from his 1950 book, because in that book, his explanation of vision was, based on the retinal image, whereas it is now (in his 1978 work), based on what (he calls) the ambient optic array (Ecological Approach, 1). Thus, I will begin by introducing the meaning of ecological optics and the ambient optic array addressed in his 1966 work,5 Husserl, Edmund, and Donn Welton. The Essential Husserl: Basic Writings in Transcendental Phenomenology (Studies in Continental Thought). Bloomington: Indiana University Press, 1999. Print. p 267. 3commenting upon his treatment of Gestalt theory, returning to his 1950 work in order to elucidate how mathematics of the difference in circumferences of two circles applied to the Pythagorean theorem can be useful for discussions of visual contours and their descriptions.Formal ontology of perceived difference in two circumferences as an arc length delivers gradients that describe different kinds of perceived surface layouts, like hollow objects, partial enclosures, places, sheets, fissures, and sticks as well as scenarios such as occluded surfaces, where a surface is blocked from view. First, a background on Gibson s laws of ecological surfaces will be developed. I will then discuss stimulus information, gradients, texture, optic information, and the ambient optic array, correlating the form of the ambient optic array to computational results found within the structure of perceived difference in circumferences.II. The Ecological Approach2.1) Ecological Optics Surface Layout, Substance, and GradientEcological optics is a theory of visual perception that takes account of the environmental context of the perceiver. Gibson introduces ecological optics with the concept of optical information, information (the kind that is sent and received), consists of messages, signs, and signals (Ecological Approach, 62). Ecological optics tries to answer the question of the kind of medium of transmission by observing the environment and using the descriptions made and differences noted to form4scientific theories of how vision and visual systems work, refining the language used for discussing the medium of perception and substance. In essence, there is a linguistic analog of the meaning of information in human communication, pictures and sculptures are apt to be displayed, and thus they contain information and make it available for anyone who looks (Ecological Approach, 63). However, the ecological approach is also, concerned with many-times-reflected light in the medium, that is, illumination, (Ecological Approach, 63). Gibson outlined several laws of ecological surfaces and their perception. He also began relating surface- gradient to the quality of a perceived surface through texture.Gibson developed a pertinent and valuable language for discussing ecological surfaces. The nine laws surface layout that Gibson gave in The Ecological Approach to Visual Perception are: 1. All persisting substances have surfaces, and all surfaces have layout.2. Any surface has resistance to deformation, depending on the viscosity ofthe substance.3. Any surface has resistance to disintegration, depending on the cohesion ofthe substance.4. Any surface has a characteristic texture, depending on the composition ofthe substance. It generally has both a layout texture and a pigmenttexture.5. Any surface has a characteristic shape, or large-scale layout.6. A surface may be strongly or weakly illuminated, in light or in shade. 57. An illuminated surface may absorb either much or little of the illumination falling on it.8. A surface has a characteristic reflectance, depending on the substance.9. A surface has a characteristic distribution of the reflectance ratios of thedifferent wavelengths of light, depending on the substance. This property is what I will call its color, in the sense that different distributions constitute different colors (Ecological Approach, 23-23)Gibson elaborates on each of the meanings of these laws. Gibson s laws are explanations for perceptions of surfaces, and are thus scientifically directed. The combination of the first and second law, the persistence of surface, and the surfaces resistance to deformation, explains why the terrestrial surface provides support to animals walking on it it is made of persisting substance that is not easily deformed upon application of force to it. The persistence of the substance necessitates a persistence of the surface layout of that substance, so it explains how layout like walls are, barriers of locomotion (Ecological Approach, 24). The second law tells us that the solidity of a surface can vary and, implies that the bog or swamp offers practically no support for standing or walking to heavy animals, and that the pond or lake offers no support (Ecological Approach, 24), and assumes that the pond is viscous to the being walking on its surface. The second law implies that, the surfaces of flexible substances are yielding or can be pushed aside, whereas the surfaces of rigid substances cannot (Ecological Approach, 25). For instance, I can6push my finger into a surface made of putty, while, by the natural order, I generally cannot push my finger into that same surface made of wood.Gibson s nine laws of surface layout mainly discuss the characteristics of the surfaces in our natural world (in the sense of the woods, the plains, the mountains and sky), but are minimalist and general enough to extend their range to surfaces that are mathematically accessible in our modern day environment, which has changed due to technology and computer interfaces. This has provided us with many surfaces that can be mathematically discovered, each of which can be characterized by combinations of the laws of surface layout provided by Gibson.I assert that, even mathematically synthesized surfaces, not seen in the non-man constructed world are still characterized by texture and contour. Gibson s laws (in combination) hold even for describing a surface, which is itself (the surface) illuminating, like a computer monitor, not just illuminated. Gibson says that, animals need to perceive what persists and what changes (Ecological Approach, 307), and that, a surface goes out of existence when its substance evaporates or disintegrates; a surface come into existence when its substance condenses or crystallizes (Ecological Approach, 307). Some day, the monitor will burn out. We also see this characteristic in the surface of water as it evaporates or turns to ice. To the touch, one might be able to say that the surface of standing water is more like the surface of ice than the surface of standing water is like the surface of the evaporated water. Each of these molecular states of the substance has a different characteristic texture to both the eye and hand.7Gibson s primary focus in outlining the laws of ecological surfaces is to provide the nomenclature for visual perception of surface layout. The fourth law addresses characteristic texture. Gibson describes texture as, the structure of a surface, as distinguished from the structure of the substance underlying the surface (Ecological Approach, 25). Gibson is talking about actual, objective surfaces. The substance underlying a surface is made of molecules, and the organization of these molecules makes a surface, the surface of a natural substance is neither homogeneous nor amorphous, but has both a chemical and a physical texture; it is generally both conglomerated and corrugated (Ecological Approach, 25). The kind of experiential texture can also describe the character of the interaction in which a change in surface layouts is perceived such that the subject experiences the environment to have differentiated qualia. Often, visually perceived surface texture is directly correlated to specific tactile textures in the natural world, and there is knowledge of the surface of a substance that is different in the tactile texture of the surface of the substance than its visual texture.There are incremental units of surfaces. Gibson says that, when referring to ecological surface texture, we are talking about the relatively fine structure of the environment at the size-level around centimeters and millimeters (Ecological Approach, 25). Gibson says that textural units are nested with larger units. For instance, pebbles, crystals, sand and grass are all part of the textured Earth. The density of the texture can be measured if the number of distinct units within a given area can be counted. The relationship between surface and texture is that textural units are made of different kinds of surfaces. On this, Gibson says that, the units of8texture vary in form, and there are forms within forms, so that the form of a texture escapes measurement (Ecological Approach, 28). We can see that certain kinds of surfaces are structurally described by the mathematics in The Geometric Pattern of Perception Theorems (Emmerson, 2009). Could this fact that the form of texture continually escapes measurement - be related to the visually textural flickering seen in The Geometric Pattern of Perception Theorems (Emmerson, 2009)? Gibson says that, the surfaces of the substances from which primitive men fashioned tools have different textures flint, clay, wood, bone, and fiber (Ecological Approach, 28). Scientific research in computational 3D architecture is just getting started, and texture, while unable to be measured, may be able to be increasingly better mathematically expressed.Contoured surfaces, nets, grids and gradients are delivered through mathematical analysis of structures, and once more, it is important to realize that smaller units are nested within larger units (Ecological Approach, 12). Mathematically delivered textural surfaces can theoretically be divided, and nested within each other, as well as having their algebraic correlates manipulated through the nesting of and substituting for variables within algebraic expressions, which deliver completely different textures. So, we see there is an analogy or metaphor within mathematical (geometric) structural analysis and ecological optics.Also, a change in surface layout is what Gibson described as motion in surface geometry. Mathematics can specify certain kinds of change in surface9information in the event of perceived motion through a change in surface layout6. In mathematical analysis, the surface of a function is what is described by the graphing of a function. The delivered surface is then imbued with attributes such as hardness and opacity, which tell us the characteristics of the synthesized substance of the object. Gibson addressed the issue of stimulus in psychology in a 1960 article, entitled The Concept of the Stimulus in Psychology 7. He says, it seems to (him) that there is a weak link in the chain of reasoning by which we explain experience and behavior, namely our concept of the stimulus (Concept of Stimulus, 694). I will try to relate the idea of stimulus to surface layout. In the article, Gibson provides accounts of historical interpretations of stimulus from Freud to Pavlov to B.F. Skinner8. B.F. Skinner said that, we frequently define the stimulus by the very doubtful property of its ability to elicit the response in question, rather than by any independent property of the stimulus itself 9. Often, in psychology, we might think of stimulus as something already perceived to which we immediately or upon deliberation react. However, Gibson critiques Skinner s remark, because, he (Skinner) suggests no remedy, however, for this doubtful scientific behavior (Concepts of Stimulus, 695). Perceived difference (in circumferences or in general)6 "Animate - Wolfram Mathematica 7 Documentation." Wolfram Mathematica 7 Documentation. N.p., n.d. Web. 7 Apr. 2010. <http://reference.wolfram.com/mathematica/ref/Animate.html>.7 Gibson, James J.. "The Concept of Stimulus in Psychology." American Psychologist 15.11 (1960): 694-703. Print. All further references to this source will be made parenthetically in the text.8 Burrhus Frederic Skinner (March 20, 1904 August 18, 1990) was an American psychologist, author, inventor, advocate for social reform, and poet."B. F. Skinner - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 9 Apr. 2010. <http://en.wikipedia.org/wiki/B.F._Skinner>.9 Skinner, B.F. Cumulative Record. New York: Appleton, Century-Crofts, 1959. p 6. 10

as arc length perceived specifies an event (of change) containing optically relevant information and provides a formal ontology by which we might get at the fundament of the independent properties of stimulus, that is, the perceptually relevant difference as objective, geometrically provable fact containing gradient of the parameters of the inherently perceived difference. It will be shown later, upon returning to Gibson s 1950 work, The Visual World, that the mathematical function describing this difference in circumferences delivers a smooth, even surface layout. This kind of eidetic, ontological, phenomenological method is a potential remedy to Skinner s remark. Gibson might support such an idea, because he says, gradients, patterns, and other invariants are not part of existing geometrical optics, but they are physical facts (Concepts of Stimulus, 701), and that what was needed for a psychophysics of visual perception was not more theorizing about cues but more attention to geometrical optics (Concepts of Stimulus, 701). There is optically, a perceived difference in circumferences as arc length perceived, and this can be studied in order to get an idea of the information within stimulus specifying change in the environment or motion within it (a change in surface layout, contour, slant, gradient, and potentially texture). Gibson says that some theorists, go further and, by arguing that an experimenter cannot define the stimulus anyway except in terms of his perception, (reach) a philosophical position of subjectivism (Concepts of Stimulus, 696). This is not what occurs in formal ontology, because, in the case of change in circumferences, it is eidetically arrived at by phenomenological description in the language of pure mathematics, and thus is universal science, 11placing objectivity in a subjective framework and giving recognition to the two ideas mutual interpenetration.The difference in circumferences equaling an arc length is an essential geometric insight, which has not yet been given enough attention in the analysis of experimental data in visual studies.

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People speak of Energy to describe the phenomenon of that which is neither created nor destroyed, but really, all that is needed to describe that phenomenon is contained within the, “phenomenological velocity,” equation, also known as V-Curvature.